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Properties

Label 8-570e4-1.1-c3e4-0-0
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.27927\times 10^{6}$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·3-s + 4·4-s + 10·5-s − 24·6-s + 36·7-s + 16·8-s + 9·9-s − 40·10-s − 22·11-s + 24·12-s + 92·13-s − 144·14-s + 60·15-s − 64·16-s − 105·17-s − 36·18-s + 136·19-s + 40·20-s + 216·21-s + 88·22-s − 56·23-s + 96·24-s + 25·25-s − 368·26-s − 54·27-s + 144·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s + 1.94·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s − 0.603·11-s + 0.577·12-s + 1.96·13-s − 2.74·14-s + 1.03·15-s − 16-s − 1.49·17-s − 0.471·18-s + 1.64·19-s + 0.447·20-s + 2.24·21-s + 0.852·22-s − 0.507·23-s + 0.816·24-s + 1/5·25-s − 2.77·26-s − 0.384·27-s + 0.971·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.27927\times 10^{6}\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3311578304\)
\(L(\frac12)\) \(\approx\) \(0.3311578304\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 136 T + 783 p T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
good7$D_{4}$ \( ( 1 - 18 T + 382 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + p T + 236 p T^{2} + p^{4} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 46 T - 81 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 105 T - 691 T^{2} + 198450 T^{3} + 58961262 T^{4} + 198450 p^{3} T^{5} - 691 p^{6} T^{6} + 105 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 56 T + 2658 T^{2} - 1335936 T^{3} - 185963117 T^{4} - 1335936 p^{3} T^{5} + 2658 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 - 153 T - 9565 T^{2} + 2418012 T^{3} - 78460746 T^{4} + 2418012 p^{3} T^{5} - 9565 p^{6} T^{6} - 153 p^{9} T^{7} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 - 222 T + 65743 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 582 T + 185602 T^{2} + 582 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 228 T - 43414 T^{2} - 9677232 T^{3} + 767884899 T^{4} - 9677232 p^{3} T^{5} - 43414 p^{6} T^{6} + 228 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 + 212 T - 119146 T^{2} + 1076112 T^{3} + 17030802443 T^{4} + 1076112 p^{3} T^{5} - 119146 p^{6} T^{6} + 212 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 + 273 T - 143953 T^{2} + 2958228 T^{3} + 29217344628 T^{4} + 2958228 p^{3} T^{5} - 143953 p^{6} T^{6} + 273 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 299 T - 138207 T^{2} + 20973654 T^{3} + 16331287858 T^{4} + 20973654 p^{3} T^{5} - 138207 p^{6} T^{6} - 299 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 453 T - 235195 T^{2} - 13429638 T^{3} + 109629530364 T^{4} - 13429638 p^{3} T^{5} - 235195 p^{6} T^{6} - 453 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 457 T - 216379 T^{2} + 13131438 T^{3} + 81893722754 T^{4} + 13131438 p^{3} T^{5} - 216379 p^{6} T^{6} - 457 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 1648 T + 1441562 T^{2} - 1108800768 T^{3} + 716645716043 T^{4} - 1108800768 p^{3} T^{5} + 1441562 p^{6} T^{6} - 1648 p^{9} T^{7} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 + 1239 T + 630425 T^{2} + 234014886 T^{3} + 140937876264 T^{4} + 234014886 p^{3} T^{5} + 630425 p^{6} T^{6} + 1239 p^{9} T^{7} + p^{12} T^{8} \)
73$D_4\times C_2$ \( 1 + 4 p T - 215126 T^{2} - 1910576 p T^{3} - 94269219677 T^{4} - 1910576 p^{4} T^{5} - 215126 p^{6} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 15 T - 80293 T^{2} - 13583400 T^{3} - 236715240972 T^{4} - 13583400 p^{3} T^{5} - 80293 p^{6} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 313 T + 1165660 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 229 T - 1139511 T^{2} - 49918794 T^{3} + 886799000014 T^{4} - 49918794 p^{3} T^{5} - 1139511 p^{6} T^{6} + 229 p^{9} T^{7} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 + 1050 T - 979606 T^{2} + 269598000 T^{3} + 2495971407807 T^{4} + 269598000 p^{3} T^{5} - 979606 p^{6} T^{6} + 1050 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.54053354164858673114938632501, −6.95822286981293337039918312090, −6.86643847129048890658662568116, −6.83627077681056343934685448502, −6.58143921318219377394820655569, −6.21015696420021900240581739986, −5.90513908129120528349470005135, −5.35030647327690482175736860261, −5.24378223244806136011236779545, −5.23833419595007916557684371139, −4.98378132437616436612615075180, −4.61535862907753880873288326729, −4.24230862582467629308043377330, −3.90491505791675420972133811775, −3.77786643214017824534297637081, −3.25302043094041052562975003617, −3.13256302532425347209460825473, −2.73632802163816079456265817738, −2.28005735691308386101124472357, −1.91493971110154579904168034785, −1.87402130329706119333328002073, −1.37012071921601830722976617882, −1.21785294547068484837626920303, −0.903270582174526713625775467691, −0.079380417476605580142406197661, 0.079380417476605580142406197661, 0.903270582174526713625775467691, 1.21785294547068484837626920303, 1.37012071921601830722976617882, 1.87402130329706119333328002073, 1.91493971110154579904168034785, 2.28005735691308386101124472357, 2.73632802163816079456265817738, 3.13256302532425347209460825473, 3.25302043094041052562975003617, 3.77786643214017824534297637081, 3.90491505791675420972133811775, 4.24230862582467629308043377330, 4.61535862907753880873288326729, 4.98378132437616436612615075180, 5.23833419595007916557684371139, 5.24378223244806136011236779545, 5.35030647327690482175736860261, 5.90513908129120528349470005135, 6.21015696420021900240581739986, 6.58143921318219377394820655569, 6.83627077681056343934685448502, 6.86643847129048890658662568116, 6.95822286981293337039918312090, 7.54053354164858673114938632501

Graph of the $Z$-function along the critical line